solving equation 7(√x/1-x) + 8(√1-x/x) = 15 following roots are obtained
To solve the equation 7(√x/1-x) + 8(√1-x/x) = 15 and find its roots, you can follow these steps:
Step 1: Simplify the equation
Multiply each term by the denominators to get rid of the fractions. This will result in a quadratic equation. The equation becomes:
7(√x) + 8(√(1-x)) = 15(1-x)
Step 2: Square both sides of the equation
By squaring both sides of the equation, we eliminate the square root symbols. This gives:
49x + 112√x(1-x) + 64(1-x) = 225(1-x)^2
Step 3: Expand and simplify
Expand the terms and simplify the equation. This will help to get rid of the remaining square roots. The equation becomes:
49x + 112√x - 112x√x + 64 - 64x = 225(1 - 2x + x^2)
Step 4: Combine like terms
Rearrange the equation to collect similar terms. This will help in solving the quadratic equation. The equation becomes:
x^2 - 105x + 49√x - 176x√x + 161x - 161 + 112√x - 64 = 0
Step 5: Express in terms of √x
Since the equation involves both x and √x terms, we can express everything in terms of √x. Replace x with (√x)^2 and simplify the equation. This leads to:
u^4 - (105+176)u^2 + (49+161)u - (161+64)=0
where u = √x
Step 6: Solve the quadratic equation
Now, you have a quadratic equation in terms of u. Use any suitable method to solve this equation. You can factorize, complete the square, or use the quadratic formula to find the roots of the equation.
Once you find the roots of the quadratic equation, substitute them back into u = √x to get the corresponding values of x. These values are the roots of the original equation.
Note: Solving the quadratic equation may lead to complex roots or additional extraneous solutions. Be sure to check the solutions obtained to verify their validity.